manuscript drafting

first draft (pre-membrane mechanics model)

brainstorming main points

  1. droplet assists CME progression through newly exposed membrane
    • we can show this because at the infinite volume limit, energy from droplet collapses on wetting component
    • the wetting component always is net negative free energy and negative slope at h=0
  2. droplet volume sets the opposing force to invagination through change in cytosol contact area, resulting in energy barrier
    • as volume decreases, the surface tension term starts to dominate
    • droplet helps late in internalization, not initial bending
      • when wetting and surface tension are combined, there will always be an energy barrier because gamma_m is always < gamma_d
  3. what sets the position and magnitude of this energy barrier?
    • droplet volume: h decreases as volume increases to infinity
    • gamma_m/gamma_d: haven't investigated this yet
  4. comparison of reasonable droplet energy magnitudes to energy scales in CME

introduction

  • some of my experimental data with actin clumps:

model description (fig. 1)

  • we just have the simple yeast version for now: cell wall allows us to have flat base membrane
    • non-cell wall version will have more complicated geometry since the droplet should bend the membrane outward
llps cme Manuscript Drafting - Page 1.png

model cartoon

  • placeholder: labeled_cartoon.png
  • current progress of 3D model (see [BROKEN LINK: *3D geometry] for code): 20210301_3d_llps_model_v1.mov
  • feature labeled snapshots of beginning, middle, and end

free energy equation

I'm keeping these up-to-date in [BROKEN LINK: *summary]

arranged by force contributions:

\begin{equation} F=\sigma\left(S_{o}+S_{m}+S_{i}\right)+\frac{K}{2} S_{i}\left(\frac{2}{R_{i}}-2C_{i}\right)^{2}+\gamma_{d} S_{d}-\gamma_{m} S_{m}-\gamma_{i} S_{i}+P V_{i} \end{equation}

arranged by shapes:

\begin{equation} F= \sigma S_{o} + \left(\sigma - \gamma_{m} \right) S_{m} + \gamma_{d} S_{d} + \frac{K}{2} S_{i} \left(\frac{2}{R_{i}} - 2 C_{i}\right)^{2} + P V_{i} + \left(\sigma - \gamma_{i} \right) S_{i} \end{equation}

free energy in terms of h and constants

\begin{equation} F(h)= \left(\gamma_{m} \alpha+\gamma_{d} \epsilon\right) \left(\frac{S_{i}^{3 / 2}}{\sqrt{\pi}}\left(\frac{3}{2} h-h^{3}\right)+3 V_{d}\right)^{2/3} - 2 P \eta h^{3}+\left(\left(\sigma-\gamma_{m}\right) S_{i}+8 K \pi\right) h^{2} +\left(3 P \eta-8 K C_{i} \sqrt{\pi S_{i}}\right) h+B \end{equation} \begin{equation} \alpha = \left(\frac{\pi\left(\frac{\gamma_{m}}{\gamma_{d}}+1\right)^{3}}{\left(\frac{\gamma_{m}}{\gamma_{d}}-1\right)\left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \epsilon = \left(\frac{8 \pi}{\left(-\frac{\gamma_{m}}{\gamma_{d}}+1\right) \left(\frac{\gamma_{m}}{\gamma_{d}}+2\right)^{2}}\right)^{1 / 3} \end{equation} \begin{equation} \eta=\frac{S_i^{3/2}}{6 \sqrt{\pi }} \end{equation} \begin{equation} B=\sigma \pi A_t^2+ S_i \left(\gamma _m-\gamma _i+2 K C_{i}^{2}\right) \end{equation}

∆ free energy equation

  • still need to re-define ∆ surface area terms to be S(h)-S(0)
  • these should allow us to see the limit at infinite volume

results (figs. 2-?)

droplet assists CME progression through newly exposed membrane

llps cme Manuscript Drafting - Page 2.png

  • [BROKEN LINK: *at very high droplet volume, exposure of new membrane to droplet dominates energy contribution from droplet]:

    labeled_cartoon.png
    \begin{equation} S_{m}=\pi\left(A_{d}^{2}-A_{i}^{2}\right) \end{equation} \begin{equation} A_{i}=\sqrt{\frac{S_{i}}{\pi} \left(1 - h^{2}\right)} \end{equation}
  • show that ∆ A_d approaches 0 at infinite V_d, which means ∆ S_d also approaches 0 (inhibitory effect of droplet)
  • because ∆ A_i (assistive effect of droplet) is independent of V_d, droplet will always be net assistive at high V_d and gamma_m < gamma_d

  • from [BROKEN LINK: *huge droplet]:

    bigdrop.png droplet_energy_contributions_bigdrop.png

  • from [BROKEN LINK: *normal droplet]:

    normal_droplet.png droplet_energy_contributions.png

droplet volume sets the opposing force to invagination through change in cytosol contact area, resulting in energy barrier

llps cme Manuscript Drafting - Page 3.png

  • from [BROKEN LINK: *contribution of droplet to $F'(h=0)$ (local stability)]:

    \begin{equation} F^{\prime}(h=0)=\frac{\left(\gamma_{m} \alpha + \gamma_{d} \epsilon \right) S_{i}^{3 / 2}}{\sqrt{\pi} \left(3 V_{d}\right)^{1 / 3}}+3 P \eta-8 K C_{i} \sqrt{\pi S_{i}} \end{equation}
    • the droplet only contributes to \(F^{\prime}(0)\) through the first term
    • this term is always positive for \(\gamma_{m} < \gamma_{d}\) (which is always true), meaning the droplet can never be assistive to the flat-curved membrane transition
    • as \(V_{d}\) gets infinitely large, this effect goes to 0
  • droplet helps late in internalization, not initial bending
    • show what parameters affect the h coordinate and ∆F magnitude at which the free energy barrier is located
  • (IDEA) at finite size, when does thermodynamics improve by having the droplet?
    • may not be able to answer
    • if we separate F into F_droplet and F_non-droplet, what value of gamma_m/gamma_d does ∆F_droplet(h=1) = 0
      • y-axis: volume of droplet
      • x-axis: gamma_m/gamma_d

droplet volume and interfacial tensions set energy barrier characteristics

llps cme Manuscript Drafting - Page 4.png
  • how does this compare to opposing energy magnitudes?

how do droplet energy magnitudes compare to other known energy contributions in CME

llps cme Manuscript Drafting - Page 5.png
  • this is where we make comparisons to physiological ranges of energy

discussion

  • initial barrier might actually be physiologically important and align with experimental observations
    • CME is defined by highly variable initiation/recruitment stage followed by fast, reproducible internalization stage
    • as the droplet grows, the energy barrier shifts to lower h

second draft

brainstorming

  • any paper I could model this after?
  • constant area vs. constant curvature models

    • cooperative curvature model (Mund et al., 2022)

      2023-01-24_23-02-31_2023_01_24_Selection_10.png

    • supporting constant area model (Sochacki et al., 2021)

      2023-01-25_09-39-29_2023_01_25_Selection_01.png

    • supporting constant curvature model (Willy et al., 2021)

      2023-01-25_09-42-33_2023_01_25_Selection_02.png

    • combined model (Tagiltsev, Haselwandter and Scheuring, 2021)

      2023-01-25_09-45-48_2023_01_25_Selection_03.png

    • heterogeneous model (“Evolving models for assembling and shaping clathrin-coated pits | Journal of Cell Biology | Rockefeller University Press,” no date)

      2023-01-25_09-47-53_2023_01_25_Selection_04.png
    • currently working here on free energy landscapes of combined toy model to overlay on contour plot like this hypothetical_2d_model.png

    • can there be a droplet energy barrier in constant curvature model? 

      #import libraries
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

#defining constant parameters
sigma = 0.001 #pN/nm - membrane tension
A_t = 2000 #nm - total plane radius
R_i = 50 #nm - invagination radius (constant curvature model)
gamma_d = 0.015 #pN/nm - droplet-cytosol surface tension
gamma_m = 0.002 #pN/nm - droplet-membrane surface tension
gamma_i = 0.002 #pN/nm - droplet-invagination surface tension
theta_d = np.arccos(gamma_m/gamma_d) #droplet-membrane contact angle (radians)
omega = 0.11 #pN/nm - polymerization energy density (pN/nm*nm^2 = pN*nm)
drop_radius = 1500 #nm - initial droplet radius used to set V_d (V_d is constant but radius changes as the vesicle grows)
V_d = 4*np.pi*drop_radius**2
P = 0.1 #pN/nm^2 - turgor pressure

mu = (3/(np.pi*(2+np.cos(theta_d))*(1-np.cos(theta_d))**2))**(1/3)
lamb = np.pi*(mu**2)*(1-(gamma_m/gamma_d)**2)
nu = (4/3)*np.pi*R_i**3
tau = 4*np.pi*R_i**2
psi = 2*np.pi*mu**2*(1-(gamma_m/gamma_d))

h = np.linspace(0,1,100)

S_m = lambda h: lamb*(V_d + nu*(3*h**2 - 2*h**3))**(2/3) - tau*(h-h**2)
S_d = lambda h : psi*(V_d + nu*(3*h**2 - 2*h**3))**(2/3)
S_i = lambda h : tau*h

#defining and plotting more changes in free energy
Sm = S_m(h)
Sd = S_d(h)
Si = S_i(h)

deltaF_surface = gamma_d*Sd - gamma_d*Sd[0]
deltaF_wetting = ((gamma_m*Sm - gamma_m*Sm[0])+(gamma_i*Si-gamma_i*Si[0]))
deltaF_condensate = deltaF_surface - deltaF_wetting

plt.figure()
plt.plot(h,deltaF_surface,'b-',label='surface tension')
plt.plot(h,deltaF_wetting,'k-',label='wetting energy')
plt.plot(h,deltaF_condensate,'g-',label='condensate energy')
plt.xlabel('h: invagination distance/vesicle height');plt.ylabel(r'$\Delta$ Energy (pN*nm)')
plt.legend()
plt.savefig('2023-01-18_condensate_energy.png',dpi=720,bbox_inches='tight')
plt.show()
      bd500eef101cdc4725baaea5926132680db9cbc9.png
      • yes; just maybe with a pretty narrow set of parameters
  • schematic cartoons

main points

  1. droplet assists CME progression through newly exposed membrane
    • true for both alternative toy models
    • is the continuum model relevant yet? maybe show a simulation initializing around the predicted energy barrier where the droplet is necessary for completion
    • what features are parameter-independent?
      • the wetting component always is net negative free energy and negative slope at h=0
    • what features are parameter-dependent?
      • energy from droplet collapses on wetting component as droplet volume approaches infinity
      • net change in energy from droplet depends on interfacial tensions, coat area, droplet volume
  2. droplet can produce energy barrier
    • always true for constant area model
    • sometimes true for constant curvature model
    • in both models, barrier gets smaller/earlier as volume increases or wetting/surface tension ratio increases
    • for continuum model, compare with/without droplet?
  3. what do the droplet parameters need to be in order to make an impact on endocytosis energetics?
    • lots of phase diagrams
  4. what do experimental measurements of these droplet parameters predict about the impact of the droplet in endocytosis?
    • combination of FRAP and merging dynamics to estimate surface tension and viscosity in cytoplasm
    • unlikely to have time for this, but set up synthetic membrane experiments with purified proteins and induced curvature

introduction

model explanation (fig. 1)

fig1_idea.png

results (figs. 2-?)

references

Akamatsu, M. et al. (2020) “Principles of self-organization and load adaptation by the actin cytoskeleton during clathrin-mediated endocytosis,” Elife. Edited by P. Bassereau et al., 9, p. e49840. Available at: https://doi.org/10.7554/eLife.49840.
Avinoam, O. et al. (2015) “Endocytic sites mature by continuous bending and remodeling of the clathrin coat,” Science, 348(6241), pp. 1369–1372. Available at: https://doi.org/10.1126/science.aaa9555.
Bergeron-Sandoval, L.-P. et al. (2021) “Endocytic proteins with prion-like domains form viscoelastic condensates that enable membrane remodeling,” Proceedings of the national academy of sciences, 118(50). Available at: https://doi.org/10.1073/pnas.2113789118.
Boulant, S. et al. (2011) “Actin dynamics counteract membrane tension during clathrin-mediated endocytosis,” Nature cell biology, 13(9), pp. 1124–1131. Available at: https://doi.org/10.1038/ncb2307.
Dmitrieff, S. and Nédélec, F. (2015) “Membrane Mechanics of Endocytosis in Cells with Turgor,” Plos computational biology, 11(10). Available at: https://doi.org/10.1371/journal.pcbi.1004538.
“Evolving models for assembling and shaping clathrin-coated pits | Journal of Cell Biology | Rockefeller University Press” (no date). https://rupress.org/jcb/article/219/9/e202005126/152007/Evolving-models-for-assembling-and-shaping.
Hassinger, J.E. et al. (2017) “Design principles for robust vesiculation in clathrin-mediated endocytosis,” Proceedings of the national academy of sciences, p. 201617705. Available at: https://doi.org/10.1073/pnas.1617705114.
Kukulski, W. et al. (2012) “Plasma Membrane Reshaping during Endocytosis Is Revealed by Time-Resolved Electron Tomography,” Cell, 150(3), pp. 508–520. Available at: https://doi.org/10.1016/j.cell.2012.05.046.
Kumar, G. and Sain, A. (2016) “Shape transitions during clathrin-induced endocytosis,” Physical review e, 94(6), p. 062404. Available at: https://doi.org/10.1103/PhysRevE.94.062404.
Mund, M. et al. (2022) “Superresolution microscopy reveals partial preassembly and subsequent bending of the clathrin coat during endocytosis.” bioRxiv, p. 2021.10.12.463947. Available at: https://doi.org/10.1101/2021.10.12.463947.
Nickaeen, M. et al. (2019) “Actin assembly produces sufficient forces for endocytosis in yeast,” Molecular biology of the cell, 30(16), pp. 2014–2024. Available at: https://doi.org/10.1091/mbc.E19-01-0059.
Sochacki, K.A. et al. (2017) “Endocytic proteins are partitioned at the edge of the clathrin lattice in mammalian cells,” Nature cell biology, 19(4), pp. 352–361. Available at: https://doi.org/10.1038/ncb3498.
Sochacki, K.A. et al. (2021) “The structure and spontaneous curvature of clathrin lattices at the plasma membrane,” Developmental cell, 0(0). Available at: https://doi.org/10.1016/j.devcel.2021.03.017.
Tagiltsev, G., Haselwandter, C.A. and Scheuring, S. (2021) “Nanodissected elastically loaded clathrin lattices relax to increased curvature,” Science advances, 7(33), p. eabg9934. Available at: https://doi.org/10.1126/sciadv.abg9934.
Willy, N.M. et al. (2021) “De novo endocytic clathrin coats develop curvature at early stages of their formation,” Developmental cell, 0(0). Available at: https://doi.org/10.1016/j.devcel.2021.10.019.
Xiao, K., Wu, C.-X. and Ma, R. (2022) “Vesiculation mechanisms mediated by anisotropic proteins.” arXiv. Available at: https://arxiv.org/abs/2211.09490.
Yarar, D., Waterman-Storer, C.M. and Schmid, S.L. (2004) “A Dynamic Actin Cytoskeleton Functions at Multiple Stages of Clathrin-mediated Endocytosis,” Molecular biology of the cell, 16(2), pp. 964–975. Available at: https://doi.org/10.1091/mbc.e04-09-0774.

Author: Max Ferrin

Created: 2023-01-26 Thu 07:44

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